@MISC{Jacob06multiplepartitions,, author = {P. Jacob and P. Mathieu}, title = { Multiple partitions, lattice paths and a . . . }, year = {2006} }

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Abstract

A bijection is presented between (1): partitions with conditions fj + fj+1 ≤ k − 1 and f1 ≤ i − 1, where fj is the frequency of the part j in the partition, and (2): sets of k − 1 ordered partitions (n (1),n (2) , · · ·,n (k−1) ) such that n (j) ℓ ≥ n(j) ℓ+1 + 2j and n(j) mj ≥ j + max(j − i+1,0)+2j(mj+1+ · · ·+mk−1), where mj is the number of parts in n (j). This bijection entails an elementary and constructive proof of the Andrews multiple-sum enumerating partitions with frequency conditions. A very natural relation between the k − 1 ordered partitions and restricted paths is also presented, which reveals our bijection to be a modification of Bressoud’s version of the Burge correspondence.